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A characterization of jump operators

Published online by Cambridge University Press:  12 March 2014

Howard Becker
Howard Becker*
Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
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Extract

The topic of this paper is jump operators, a subject which originated with some questions of Martin and a partial answer to them obtained by Steel [18]. The topic of jump operators is a part of the general study of the structure of the Turing degrees, but it is concerned with an aspect of that structure which is different from the usual concerns of classical recursion theory. Specifically, it is concerned with studying functions on the degrees, such as the Turing jump operator, the hyperjump operator, and the sharp operator.

Roughly speaking, a jump operator is a definable ≤T-increasing function on the Turing degrees. The purpose of this paper is to characterize the jump operators, in terms of concepts from descriptive set theory. Again roughly speaking, the main theorem states that all jump operators (other than the identity function) are obtained from pointclasses by the same process by which the hyperjump operator is obtained from the pointclass Π11; that is, if Γ is the pointclass, then the operator maps the real x to the universal Γ(x) subset of ω. This characterization theorem has some corollaries, one of which answers a question of Steel [18]. In §1 we give a brief introduction to this general topic, followed by a brief (and still somewhat imprecise) description of the results contained in this paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 708 - 728
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1]Becker, H., Inner model operators and the continuum hypothesis, Proceedings of the American Mathematical Society, vol. 96 (1986), pp. 126–129.CrossRefGoogle Scholar
[2]Jech, T. J., Set theory, Academic Press, New York, 1978.Google Scholar
[3]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229–308.CrossRefGoogle Scholar
[4]Kechris, A. S., The theory of countable analytical sets, Transactions of the American Mathematical Society, vol. 202 (1975), pp. 259–297.CrossRefGoogle Scholar
[5]Kechris, A. S., “AD + uniformization” is equivalent to “half ADR”, Cabal Seminar 81–85, Lecture Notes in Mathematics, Springer-Verlag, Berlin (to appear).CrossRefGoogle Scholar
[6]Kechris, A. S. and Moschovakis, Y. N. (editors), Cabal seminar 76–77, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, Berlin, 1978.CrossRefGoogle Scholar
[7]Kechris, A. S., Solovay, R. M., and Steel, J. R., The axiom of determinacy and the prewellordering property, Cabal seminar 77–79 (Kechris, A. S.et al., editors), Lecture Notes in Mathematics, vol. 839, Springer-Verlag, Berlin, 1981, pp. 101–125.CrossRefGoogle Scholar
[8]Lachlan, A. H., Uniform enumeration operations, this Journal, vol. 40 (1975), pp. 401–409.Google Scholar
[9]Martin, D. A., The axiom of determinateness and reduction principles in the analytical hierarchy, Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 687–689.CrossRefGoogle Scholar
[10]Martin, D. A., The largest countable this, that, and the other, Cabal seminar 79–81 (Kechris, A. S.et al., editors), Lecture Notes in Mathematics, vol. 1019, Springer-Verlag, Berlin, 1983, pp. 97–106.CrossRefGoogle Scholar
[11]Moschovakis, Y. N., Hyperanalytic predicates, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 249–282.CrossRefGoogle Scholar
[12]Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[13]Mycielski, J. and Świerczkowski, S., On the Lebesgue measurability and the axiom of determinateness, Fundamenta Mathematical vol. 54 (1964), pp. 67–71.CrossRefGoogle Scholar
[14]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[15]Sacks, G. E., On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140–141.CrossRefGoogle Scholar
[16]Slaman, T. A. and Steel, J. R., Definable functions on degrees, Cabal seminar 81–85, Lecture Notes in Mathematics, Springer-Verlag, Berlin (to appear).Google Scholar
[17]Steel, J. R., Closure properties of pointclasses, Cabal seminar 77–79 (Kechris, A. S.et al., editors), Lecture Notes in Mathematics, vol. 839, Springer-Verlag, Berlin, 1981, pp. 147–163.CrossRefGoogle Scholar
[18]Steel, J. R., A classification of jump operators, this Journal, vol. 47 (1982), pp. 347–358.Google Scholar
[19]van Wesep, R., Wadge degrees and descriptive set theory, in [6], pp. 151–170.CrossRefGoogle Scholar